Section 5.1 (Homework) Consider the following Use six rectangles to find estimates of each type for the area under the given graph of f from x = 0 to x = 12.

a) i)

$\Delta X$ = b-a/n = 12-0/6 = 2

L6 = $\Delta X (f(x_0) + f(x_2) + f(x_4) + f(x_6) + f(x_8) + f(x_{10}))$

= 2(9 + 8.7 + 8.15 + 7.15 + 6 + 4) = 86

ii)

R6 = $\Delta X (f(x_2) + f(x_4) + f(x_6) + f(x_8) + f(x_{10}) + f(x_{12}))$ = 2(8.7 + 8.15 + 7.15 + 6 + 4+ 1) = 70

iii)

M6 = $\Delta X (f(x_1) + f(x_3) + f(x_5) + f(x_7) + f(x_{9}) + f(x_{11}))$ = 2(8.9 + 8.6 + 7.85 + 6.5 + 5.25+ 2.9) = 80

b) L6 is overestimate.

c) R6 is underestimate.

d) M6 gives the best estimate.